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Augmented fifth
Augmented fifth
Augmented fifth
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augmented fifth
Inversediminished fourth
Name
Other names-
AbbreviationA5[1]
Size
Semitones8
Interval class4
Just interval25:16,[2] 11:7, 6561:4096
Cents
12-Tone equal temperament800
Just intonation773, 782.5, 816
Augmented fifth on C.

In Western classical music, an augmented fifth (Playⓘ) is an interval produced by widening a perfect fifth by a chromatic semitone.[1][3] For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C♭ to G, and from C to G♯ are augmented fifths, spanning eight semitones. Being augmented, it is considered a dissonant interval.[4]

Its inversion is the diminished fourth, and its enharmonic equivalent is the minor sixth.

The augmented fifth only began to make an appearance at the beginning of the common practice period of music as a consequence of composers seeking to strengthen the normally weak seventh degree when composing music in minor modes.

This was achieved by chromatically raising the seventh degree (or subtonic) to match that of the unstable seventh degree (or leading tone) of the major mode (an increasingly widespread practice that led to the creation of a modified version of the minor scale known as the harmonic minor scale).

A consequence of this was that the interval between the minor mode's already lowered third degree (mediant) and the newly raised seventh degree (leading note), previously a perfect fifth, had now been "augmented" by a semitone.

Another result of this practice was the appearance of the first augmented triads, built on the same (mediant) degree, in place of the naturally occurring major chord.

As music became increasingly chromatic, the augmented fifth was used with correspondingly greater freedom and also became a common component of jazz chords. Near the end of the nineteenth century the augmented fifth became commonly used in a dominant chord. This would create an augmented dominant (or V) chord. The augmented fifth of the chord would then act as a leading tone to the third of the next chord. This augmented V chord would never precede a minor tonic (or i) chord since the augmented fifth of the dominant chord is identical to the third of the tonic chord.

In an equal tempered tuning, an augmented fifth is equal to eight semitones, a ratio of 22/3:1 (about 1.587:1), or 800 cents. The 25:16 just augmented fifth arises in the C harmonic minor scale between E♭ and B.[5] Playⓘ

The augmented fifth is a context-dependent dissonance. That is, when heard in certain contexts, such as that described above, the interval will sound dissonant. In other contexts, however, the same eight-semitone interval will simply be heard (and notated) as its consonant enharmonic equivalent, the minor sixth.

An augmented fifth in equal temperament
An augmented fifth in equal temperament
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An augmented fifth in just intonation
An augmented fifth in just intonation
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Pythagorean augmented fifth

[edit]

The Pythagorean augmented fifth is the ratio 6561:4096, or about 815.64 cents.[6]

See also

[edit]

References

[edit]
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Augmented fifth

View on Grokipedia
from Grokipedia
In Western music theory, the augmented fifth is an interval spanning eight semitones, formed by widening a perfect fifth (seven semitones) by one chromatic semitone. This can be achieved by raising the upper note or lowering the lower note of a perfect fifth by a half step, such as from F to C♯ or from F♭ to C. The augmented fifth is enharmonically equivalent to a minor sixth, sharing the same pitch distance but differing in contextual naming and function within harmonic progressions. When inverted, it becomes a diminished fourth, maintaining the interval's dissonant character. Due to its instability, the augmented fifth is classified as a dissonance in tonal music, often requiring resolution to a consonance like a or for harmonic stability. In harmonic contexts, the augmented fifth serves as the upper interval in an augmented triad, which comprises a , , and augmented fifth—effectively two stacked vertically. For example, a G-augmented triad includes the notes G, B, and D♯, where the augmented fifth spans from G to D♯. Augmented triads, and thus the augmented fifth, are inherently chromatic and non-diatonic, introducing tension and color in compositions across classical, , and modern genres, though they appear infrequently in strictly tonal frameworks due to their unresolved quality.

Definition and Properties

Core Definition

The augmented fifth is a musical interval formed by enlarging a by one chromatic , creating a dissonant quality that distinguishes it from more intervals. This alteration raises the upper note of the , such as shifting from G to G♯ above C, producing an interval that spans five letter names but with heightened tension due to the chromatic adjustment. In standard notation, the augmented fifth appears as the distance from C to G♯ or from F to C♯, and it is abbreviated as A5 or +5. Within pitch-class , this interval is enharmonically equivalent to a minor sixth, sharing the same intervallic content in atonal contexts, though its directional and contextual use in tonal music emphasizes its fifth-like identity. As a key component of harmony, the augmented fifth functions as the upper interval in an augmented triad, exemplified by the notes C, E, and G♯, where it pairs with a major third to form a symmetrical, fully augmented structure. This configuration contributes to the triad's overall instability, amplifying the interval's inherent dissonance. Auditorily, the augmented fifth is perceived as tense and unstable, evoking a sense of urgency that typically resolves outward to a perfect fifth or inward to an octave for consonant closure.

Acoustic Properties

The augmented fifth interval derives its dissonant character primarily from the acoustic interaction of its component tones' harmonics, stemming from a more complex ratio compared to consonant intervals like the . In , the augmented fifth corresponds to a ratio of 8:5 (approximately 1.6), which involves higher prime factors than the 's simpler 3:2 ratio (approximately 1.5). This complexity results in poorer alignment of the harmonics—integer multiples of the fundamental —leading to greater perceptual interference and reduced . When the augmented fifth is played, the nearby partials of the two tones generate beat frequencies, manifesting as amplitude fluctuations that the auditory system perceives as roughness or harshness. For instance, with a lower tone at frequency ff and the upper at 85f\frac{8}{5}f, the fifth harmonic of the lower tone (5f5f) lies close to the third harmonic of the upper tone (245f≈4.8f\frac{24}{5}f \approx 4.8f), producing a beat frequency of approximately 0.2f0.2f; similar interactions occur among other partials. These beats, occurring within the range of 20–200 Hz where the ear is most sensitive to temporal variations, amplify the sensation of instability and tension, contrasting with the smoother coalescence in simpler-ratio intervals. Psychoacoustic models quantify this dissonance through measures of roughness based on critical bandwidths, which represent the frequency range over which sounds interfere perceptibly. Building on Plomp and Levelt's sensory dissonance curve, these models calculate roughness by summing contributions from pairs of partials whose frequencies fall within a , yielding higher dissonance for the augmented fifth than for the due to greater partial misalignment. Sethares extends this to complex timbres by aggregating pairwise contributions, confirming the augmented fifth's moderate-to-high roughness in harmonic spectra. In harmonic contexts, the heightens tension by clashing with surrounding tones; for example, in an unresolved such as C–E–G♯, the 8:5 ratio between C and G♯ introduces roughness that destabilizes the sonority, often evoking a need for resolution to more configurations, as the beats disrupt fusion. This perceptual effect arises independently of cultural context, rooted in universal auditory of proximity.

Interval Size and Measurement

In Equal Temperament

In 12-tone (12-TET), the augmented fifth is measured as exactly 8 semitones, corresponding to 800 cents.<grok:render type="render_inline_citation"> 51 </grok:render> This size arises from the equal division of the into 12 semitones, each spanning 100 cents, with the cent value derived from the standard cents = 1200 \times \log_2(f_2 / f_1), where f_2 and f_1 represent the frequencies of the upper and lower pitches, respectively.<grok:render type="render_inline_citation"> 51 </grok:render> A representative calculation uses frequencies with A4 at 440 Hz: the augmented fifth from C4 (261.63 Hz) to G♯4 (415.30 Hz) yields a precise of 2^{8/12} \approx 1.5874.<grok:render type="render_inline_citation"> 52 </grok:render> This reflects the geometric progression inherent to 12-TET, where each multiplies the by 2^{1/12}. Within 12-TET, the augmented fifth exhibits neutrality relative to the , as both intervals are enharmonically identical at 8 semitones, eliminating comma-based distinctions that appear in unequal tunings.<grok:render type="render_inline_citation"> 50 </grok:render> This equivalence simplifies harmonic substitutions but can introduce subtle dissonant tensions in context, akin to acoustic beating patterns observed in compound intervals. On fixed-pitch keyboard instruments like the piano, tuned to 12-TET, the augmented fifth is practically realized by spanning a white key to the adjacent black key in the layout, such as from C to G♯, facilitating its use in modern compositions without intonation adjustments.<grok:render type="render_inline_citation"> 50 </grok:render>

In Just Intonation

In just intonation, the augmented fifth is most commonly expressed by the frequency ratio 2516\frac{25}{16}, equivalent to approximately 772.63 cents. This interval arises from the product of two just major thirds, each with a ratio of 54\frac{5}{4}: 54×54=2516\frac{5}{4} \times \frac{5}{4} = \frac{25}{16}. Depending on the prime limit and scale context, alternative just ratios for the augmented fifth include 12881\frac{128}{81} (approximately 792.18 cents) in 3-limit approximations. These variations allow for closer approximations to other tuning ideals while maintaining rational frequency proportions. A representative example occurs in the just-intoned harmonic minor scale, such as from E♭ to B in C harmonic minor. With E♭ tuned to 65\frac{6}{5} and B to 158\frac{15}{8} relative to the tonic C at 1/1, the interval ratio is 15/86/5=158×56=2516\frac{15/8}{6/5} = \frac{15}{8} \times \frac{5}{6} = \frac{25}{16}. This direct stacking aligns with the scale's emphasis on pure thirds and the leading tone. These just ratios produce a context-dependent dissonance characterized by the absence of beats, as the integer proportions ensure commensurate for a purer, less rough sound compared to tempered approximations like the 12-TET augmented fifth of 800 cents. The resulting interval enhances harmonic tension in chords without the acoustic interference of incommensurate frequencies.

Tuning Systems

Pythagorean Tuning

In , the augmented fifth is derived by stacking eight pure perfect fifths of ratio 3:2 and descending four to fit within a single octave, yielding the frequency ratio (32)8/24=38212=65614096\left( \frac{3}{2} \right)^8 / 2^4 = \frac{3^8}{2^{12}} = \frac{6561}{4096}. This ratio simplifies to approximately 1.6018 and measures about 815.64 cents. The , a small interval of approximately 23.46 cents arising from the discrepancy between twelve stacked fifths and seven octaves (ratio 312219\frac{3^{12}}{2^{19}}), contributes to the augmented fifth's size exceeding certain approximations, such as the 25:16 ratio (about 772.8 cents) used in some augmented triads. This widening imparts a characteristic sharpness, often resulting in heightened dissonance or a "wolf" interval quality when juxtaposed with purer consonances in the system. In the Pythagorean scale, the augmented fifth appears as the interval from C to G♯, equivalent to two ditones (two major thirds, each 81:64) or four Pythagorean whole tones (each 9:8) in certain constructions, though its primary derivation stems from the fifths-based generator. This tuning, foundational to music theory, integrated such intervals into modal structures like the Dorian or Phrygian. Acoustically, the augmented fifth's elevated cent value of 815.64—compared to the equal-tempered 800 cents—amplifies beating and due to the overabundance of odd harmonics from the powers-of-3 numerator, enhancing its tense, unstable character in Pythagorean contexts.

Meantone and Other Historical Systems

In quarter-comma , a system prevalent from the 16th to 18th centuries that tempers the flat by one-quarter of the (approximately 5.38 cents) to achieve pure major thirds of 386.31 cents, the augmented fifth measures about 772.6 cents. This size arises from stacking eight such tempered fifths and reducing by four octaves, yielding the just interval ratio of 25:16 and purifying the augmented fifth alongside the thirds. The tempering effect reduces the augmented fifth from its Pythagorean baseline of approximately 816 cents—derived from eight pure fifths of 702 cents—by two full syntonic commas (81:80, or 21.51 cents each), distributing the comma across the chain to favor consonance in diatonic progressions while creating a wolf fifth elsewhere. In 17th-century French organ tunings, as described by Marin Mersenne, this quarter-comma distribution enabled composers to incorporate augmented fifths in chromatic passages, such as ascending lines from tonic to sharpened dominant, without venturing into the dissonant wolf interval between G♯ and E♭. Later historical systems, such as Andreas Werckmeister's III temperament from around 1681, introduced irregular tempering of select fifths (four narrowed by one-quarter of the of 23.46 cents) to enhance modulation across all keys, resulting in an augmented fifth of approximately 792 cents for greater versatility in repertoire. Similar variations appear in other well-temperaments, like those implied in Johann Sebastian Bach's Well-Tempered Clavier (circa ), where the interval hovers near 800 cents, balancing the reductions to minimize harshness in remote keys while preserving usability in chromatic contexts.

Relations to Other Intervals

Enharmonic and Inversion Equivalents

The augmented fifth is enharmonically equivalent to the , as both intervals encompass 8 semitones and produce the same pitch content despite differing spellings. For instance, the interval from C to G♯ constitutes an augmented fifth, while the same pitches notated as C to A♭ form a minor sixth; this equivalence arises because G♯ and A♭ are enharmonic notes, but the notation choice influences , such as resolving differently in chord progressions. In terms of inversion, the augmented fifth complements to a diminished fourth, following the principle that augmented intervals invert to diminished ones, with their size numbers summing to 9 (a 5th inverts to a 4th). Inverting the augmented fifth from C to G♯, for example, swaps the notes to yield G♯ to C (ascending), which spans 4 s—a diminished fourth one semitone smaller than a . Within atonal theory, the augmented fifth is classified under interval class 4, determined by taking the minimum distance between the interval and its complement (min(8, 12-8) = 4), emphasizing its equivalence to other 4-semitone spans regardless of direction or . This classification highlights structural similarities in pitch-class sets, abstracting away tonal context. Notation for the augmented fifth can introduce ambiguities, particularly in keys requiring multiple accidentals to maintain diatonic relationships. For example, an augmented fifth from C♯ might be written as C♯ to G𝄪 (G double sharp) to align with the key's scale degrees, though this is enharmonically identical to C♯ to A; such double-sharps or double-flats ensure consistency in analysis but can complicate reading without altering the sounded interval.

Comparison to Perfect Fifth

The perfect fifth encompasses seven semitones, corresponding to 700 cents in twelve-tone equal temperament (12-TET), and arises from a frequency ratio of 3:2, making it a foundational consonant interval in Western harmony. The augmented fifth is derived by raising the upper note of this perfect fifth by one semitone, yielding eight semitones or 800 cents in 12-TET. In terms of function, the is highly and stable, serving as a pillar of resolution due to its pure structure, while the augmented fifth is dissonant and tension-inducing, often acting as a leading interval in altered dominants like the V+ chord to propel chromatic motion toward resolution. On the staff, the from C to G spans five diatonic steps with natural notes, whereas the augmented fifth from C to G♯ alters the G to G♯, visually emphasizing the half-step elevation through the accidental. Theoretically, the process of augmentation aligns with Jean-Philippe Rameau's foundational principles in Traité de l'harmonie (), where chords are built from a perfect major or minor triad and modified by raising the fifth to generate dissonant structures for dramatic effect within the fundamental bass framework.

Historical and Theoretical Context

Origins in Western Music Theory

During the Renaissance, the augmented fifth gained more formal acknowledgment through counterpoint practices involving musica ficta, where performers applied unwritten accidentals to alter pitches, creating augmented intervals for harmonic correction or color. Theorists such as Gioseffo Zarlino in his Le Istitutioni harmoniche (1558) explicitly described diminished and augmented intervals, including the augmented fifth, as resulting from raising or lowering tones of perfect intervals via accidentals, classifying them as dissonances to be used sparingly for aesthetic effect in polyphony. Zarlino noted that such intervals fell outside consonant proportions derived from the senario (the numbers 1 through 6), emphasizing their secondary role in composition to avoid harshness while enabling chromatic variety in modal frameworks. The 18th-century shift toward tonal harmony further codified the augmented fifth, distinguishing it clearly from the perfect fifth in systematic harmonic theory. Jean-Philippe Rameau, in his Traité de l'harmonie (1722), analyzed the augmented fifth as the defining feature of the augmented triad, initially deeming it "worthless" for lacking a perfect fifth essential to stable chords, yet later integrating it as the major triad on the mediant degree to accommodate chromatic progressions in emerging tonal music. This period's transition from modal to tonal systems (circa 1600–1750) facilitated greater chromaticism, allowing augmented intervals like the fifth to function as passing dissonances or modulatory tools, as theorists adapted Renaissance practices to support functional harmony and key centers.

Role in Harmonic Minor Scale

In the harmonic minor scale, the augmented fifth manifests as the interval between the third and seventh degrees, spanning eight semitones. For example, in A harmonic minor (A-B-C-D-E-F-G♯-A), the notes C (third degree) and G♯ (seventh degree) form this interval. This augmented fifth is prominently featured in the built on the (III+ chord), comprising the third degree as , a major third above it, and the augmented fifth to the seventh degree—representing the sole diatonic in the major-minor tonal system without chromatic alteration. The III+ chord shares two common tones with both the tonic (i) and dominant (V) triads, differing by just one from each, which underscores its integrative role within the scale's framework. The primary purpose of incorporating the augmented fifth via the raised seventh degree lies in establishing a leading tone that bolsters the dominant function of the V chord, promoting strong resolution to the tonic and mimicking the major mode's cadential pull. This structural choice in the harmonic minor, adopted during the , enables the V chord to function as a major triad (e.g., E-G♯-B in ), heightening tension and facilitating authentic cadences in minor keys. The III+ chord, containing the augmented fifth, further supports this by implying dominant-like qualities through its proximity to V, often serving as a passing that amplifies the scale's overall dissonant drive toward resolution. In , the augmented fifth between the third and seventh degrees yields a frequency ratio of 25:16, producing a dissonant sonority that exceeds the (3:2) and intensifies the urge for tonic resolution. This ratio arises naturally from the tuning of the , where the third degree (e.g., 6/5 relative to the root) and raised seventh (15/8) combine to form the interval, contributing to the scale's characteristic tension. Nineteenth-century theorists debated the augmented fifth's "unnaturalness" within minor keys, often critiquing the augmented triad's instability as disruptive to smooth and harmonic consonance. Analysts like Gottfried Weber characterized it as "monstrous" for lacking a , rendering it incomplete and awkward in tonal contexts, though later figures such as Carl Friedrich Weitzmann defended its symmetrical potential and resolutions. This discourse highlights the interval's role as a context-dependent dissonance, integral yet challenging to the harmonic minor's design.

Applications in Composition

In Classical and Romantic Music

In the Classical period, employed the augmented fifth within to build tension in harmonic progressions, notably in the first movement of his Symphony No. 3 "Eroica," Op. 55, where an A♭ augmented triad appears at the start of the development section (around measure 247), resolving to with A♭ moving to G and E natural to F, heightening dramatic intensity before resolution to a stable . This usage exemplifies the interval's role in dominant augmentation, foreshadowing Romantic expansions of harmony, using the augmented fifth to disrupt tonal expectations and propel structural momentum. During the Romantic era, the augmented fifth's prevalence increased with the era's chromatic expansion, enabling composers to evoke ambiguity and emotional depth through and related structures; for instance, integrated the as a motivic element in the (1870), where it unifies foreground melodies and middleground harmonies as the work's primary dissonant sonority, often resolving its raised fifth to facilitate seamless modulations and symbolic tension. In Frédéric Chopin's in , Op. 55 No. 1 (c. 1843), an appears near the return to the tonic, its augmented fifth resolving to the of the dominant chord, enhancing the piece's lyrical modulation and providing a poignant release amid chromatic flourishes. This resolution pattern—augmented fifth to —became a staple in Romantic modulations, as seen in Chopin's broader oeuvre, where it underscores shifts between related keys for expressive contrast. The augmented fifth also features prominently in augmented sixth chords, which, though named for their signature interval of ten semitones, incorporate related chromatic tensions; the Italian sixth chord in C major (A♭–C–F♯) exemplifies this, functioning as a pre-dominant with its outer voices forming an augmented sixth (enharmonically equivalent to a dominant seventh's leading tone-root motion), often resolving to the V chord () while the F♯ (raised fourth) ascends to G, creating voice-leading parallels to augmented fifth resolutions in triads. Wagner's (F–B–D♯–G♯, from , 1859) builds on this tradition as a French augmented sixth, prolonging dissonance through its half-diminished voicing but evoking ambiguity in its major thirds, delaying resolution to heighten yearning. By the mid-19th century, such formations were ubiquitous in symphonic and works, reflecting the era's shift toward freer , as in Wagner's (1845), where underscore supernatural scenes.

In Jazz and Contemporary Genres

In , the augmented fifth serves as an altered tension in dominant seventh chords, denoted as 7#5 or 7+5, creating heightened dissonance and color for and resolution. For instance, the C7#5 chord comprises the notes C, E, G♯, and B♭, where the G♯ replaces the (G) to introduce instability that often resolves to a major or minor triad. This voicing appears prominently in Thelonious Monk's composition "Skippy," where a D7+5 chord adds characteristic angularity and tension to the tune's structure, exemplifying Monk's preference for dissonant extensions in his voicings. In jazz lead sheets, the augmented fifth is notated using symbols such as "#5," "+5," or simply "+" for the chord, facilitating quick reading during performances. A classic example occurs in the standard "" by , where a C+7 (C E G♯ B♭) appears at the end of the bridge in measure 24, functioning as an augmented dominant to pivot back to Fm7 and heighten the emotional arc before returning to the tonic. This substitution leverages the augmented fifth's enharmonic flexibility, allowing it to imply related dominants like E7♭9 or A♭7♯9 for varied improvisational approaches. Beyond jazz, the augmented fifth features in contemporary genres through its integral role in the , a hexatonic collection of whole steps that inherently produces augmented triads containing the interval (e.g., C-E-G♯). This scale's ambiguous, floating quality—lacking a strong tonic—lends itself to impressionistic and contexts, extending Debussy's classical precedents into modern idioms. In King Crimson's , employs whole-tone progressions in tracks like "" from the 1974 album , where the scale's augmented fifths contribute to the piece's disorienting, angular riffs and polyrhythmic intensity. In modern electronic music and film scoring, the augmented fifth enhances dissonance in sustained synth pads and harmonic clusters, evoking unease or supernatural tension without traditional resolution. Composers like integrate such intervals in orchestral motifs to amplify dramatic effect, drawing on the augmented scale's mysterious prevalent in Hollywood scores; for example, the Imperial March's chromatic distortions incorporate augmented seconds and related dissonances to underscore imperial menace, influencing electronic adaptations in synth-heavy remixes and sound design.

References

  1. https://human.libretexts.org/Courses/Prince_Georges_Community_College/PGCC_Open_Music_Theory-Fundamentals/04%253A_Intervals
  2. https://milnepublishing.geneseo.edu/fundamentals-function-form/chapter/13-triads/
  3. https://musictheory.pugetsound.edu/mt21c/AugmentedAndDiminishedIntervals.html
  4. https://pressbooks.nebraska.edu/openmusictheory/chapter/intervals/
  5. https://musictheory.pugetsound.edu/mt21c/SetTheorySection.html
  6. https://www.iconcollective.edu/basic-music-theory
  7. https://production.cbts.edu/fetch.php/71UVhU/897332/Major_Minor_Augmented_And_Diminished_Triads.pdf
  8. https://www.fittonmusic.com/writing/melody/tuningandinterval/justintonation.html
  9. https://pubs.aip.org/asa/jasa/article/142/6/3748/695137/Calculation-of-the-acoustical-properties-of
  10. https://sethares.engr.wisc.edu/consemi.html
  11. https://pmc.ncbi.nlm.nih.gov/articles/PMC6355932/
  12. https://sethares.engr.wisc.edu/paperspdf/HelmTTSS.pdf
  13. https://pmc.ncbi.nlm.nih.gov/articles/PMC10511120/
  14. https://www.kylegann.com/tuning.html
  15. https://www.allthingscello.org/wp-content/uploads/2022/11/Cello-Standards-Intonation-Just-Intonation-Interval-Ratios-1.pdf
  16. https://www.dolmetsch.com/musictheory27.htm
  17. https://www.sfu.ca/sonic-studio-webdav/handbook/Just_Tuning.html
  18. http://www.medieval.org/emfaq/harmony/pyth4.html
  19. http://www.tonalsoft.com/enc/p/pythagorean-comma.aspx
  20. https://www.mtosmt.org/issues/mto.98.4.4/mto.98.4.4.scholtz.html
  21. http://www.tonalsoft.com/enc/number/1-4cmt.aspx
  22. https://www.academia.edu/1504076/Mersenne_on_keyboard_tuning
  23. https://ccrma.stanford.edu/~mromaine/220a/fp/different-tunings.html
  24. http://www.microtonal-synthesis.com/scale_werckmeisterIII.html
  25. https://www.musictheoryacademy.com/understanding-music/enharmonic-equivalents/
  26. https://music.stackexchange.com/questions/75991/interval-analysis-augmented-5th-or-a-6th
  27. https://viva.pressbooks.pub/openmusictheory/chapter/intervals/
  28. https://www.essential-music-theory.com/inverted-intervals.html
  29. https://robertkelleyphd.com/home/atnltrms.htm
  30. https://music.stackexchange.com/questions/28804/augmented-5th-on-b-double-sharp
  31. https://mymusictheory.com/pitch/double-sharps-flats-and-enharmonics/
  32. https://www3.nd.edu/~ablachly/MUS20101/20101JustIntonationFrame.html
  33. https://music.stackexchange.com/questions/28870/what-is-a-perfect-fifth
  34. https://www.musicalinfo.web.illinois.edu/Notes/tuning.pdf
  35. https://casfaculty.case.edu/ross-duffin/baroque-ensemble-tuning/
  36. https://www.learnjazzstandards.com/blog/augmented-chord/
  37. https://music.stackexchange.com/questions/129923/difference-between-perfect-4th-and-perfect-5th
  38. https://oldmusictheory.com/harmony-2/part-ii-dissonant-chords/17-altered-chords/17-2-augmented-fifth-chords/
  39. https://www.musictheoryacademy.com/understanding-music/intervals/perfect-fifth/
  40. https://search.proquest.com/openview/a68d0fa7a6c2140da612498e14653b65/1?pq-origsite=gscholar&cbl=18750&diss=y
  41. https://music.arts.uci.edu/abauer/5.2/readings/Lester_Compositional_Theory_Eighteenth-Century_Ch_4.pdf
  42. https://symposium.music.org/19/item/1855-sixteenth-century-conception-of-harmony.html
  43. https://www.ias.edu/ideas/2015/clark-music
  44. https://viva.pressbooks.pub/openmusictheory/chapter/augmented-options/
  45. https://milnepublishing.geneseo.edu/fundamentals-function-form/chapter/16-minor-scale-variants/
  46. https://music.stackexchange.com/questions/61857/augmented-triads-in-classical-music
  47. https://www.gresham.ac.uk/watch-now/diminished-augmented
  48. https://www.jstor.org/stable/854170
  49. https://www.reddit.com/r/musictheory/comments/qr5w9x/looking_for_examples_of_augmented_chords_in_music/
  50. https://sjaakdouma.com/writings/a-modern-approach-to-chopins-augmented-modulations
  51. https://musictheory.pugetsound.edu/mt21c/TypesOfAugmentedSixthChords.html
  52. https://mtosmt.org/issues/mto.95.1.1/mto.95.1.1.rothgeb.html
  53. https://www.gresham.ac.uk/sites/default/files/transcript/2023-02-09-1800_FROLOVAWALKER_Chords-T.pdf
  54. https://bobbysternjazz.com/blog-b-natural/monks-skippy-the-etude
  55. https://www.jazzguitar.be/blog/augmented-major-seventh-chords/
  56. http://brunojazz.com/vt-AugmentedDominant.htm
  57. https://www.renhimself.com/technique/the-whole-tone-scale/
  58. https://www.youtube.com/watch?v=uqQ4aEeYLes
  59. http://whythissongisgreat.blogspot.com/2013/11/john-williams-imperial-march-on.html
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